Age-Period-Cohort-Interaction model (Luo & Hodges, 2020) across 6 US surveys
Author
Christine Lucille Kuryla
Published
February 10, 2026
Show code
library(tidyverse)library(here)library(APCI)library(patchwork)library(knitr)library(kableExtra)is_html <- knitr::is_html_output()# Helper: apply kable_styling only for HTML outputstyled_kable <-function(k, ...) {if (is_html) kable_styling(k, ...) else k}source(here::here("R", "paths.R"))source(here::here("R", "functions", "theme_srh.R"))source(here::here("R", "functions", "plot_utils.R"))source(here::here("R", "functions", "apci_analysis.R"))apci_output_dir <- here::here("output", "apc", "apci")# Survey order and labelssurveys <-c("gss", "nhanes", "meps", "nhis", "cps", "brfss")
1 Overview
This report presents an APCI (Age-Period-Cohort-Interaction) analysis of self-rated health (SRH) across six US surveys. The APCI model (Luo & Hodges, 2020) provides a third APC decomposition method, complementing the median polish (non-parametric descriptive) and BHAPC (Bayesian hierarchical parametric) approaches.
1.1 What Makes APCI Different
The APCI model reframes the classic APC identification problem. Instead of trying to separately estimate three perfectly collinear effects (age + period + cohort), it asks:
“Do period effects vary by age group?”
If yes, that variation is the cohort effect. This maps directly onto Ryder’s (1965) theory that cohort differentiation arises when historical changes affect age groups differentially.
1.2 Key Outputs
Output
Meaning
Age main effects
Expected SRH by age group, averaging over periods
Period main effects
Expected SRH by period, averaging over age groups
Cohort averages (inter-cohort)
Which birth cohorts deviate from age+period main effects?
Cohort slopes (intra-cohort)
Do cohort differences accumulate, diminish, or stay constant with age?
1.3 Convergence Context
The SRH convergence phenomenon operates from both directions: younger ages report lower SRH over time while older ages report better SRH. If this has a cohort component, we expect:
Negative cohort averages for recent birth cohorts (health worse than age+period predict)
Positive cohort averages for older cohorts (health better than expected)
The intra-cohort slopes reveal whether these differences accumulate, diminish, or stay constant
1.4 Surveys
SRH is coded higher = better. All surveys use weights via the weight argument in apci().
Survey
SRH Scale
Design Note
GSS
1–4
Weights only
NHANES
1–5
Weights only (APCI uses glm, not survey design)
MEPS
1–5
Weights only (APCI uses glm, not survey design)
NHIS
1–5
Weights only (APCI uses glm, not survey design)
CPS
1–5
Weights only
BRFSS
1–5
Weights only
1.5 Limitation: Survey Design
The APCI package uses glm() internally with a weights argument. This produces survey-weighted point estimates but standard errors do not account for complex survey design (strata/PSU clustering). Point estimates are correct; inference (p-values, CIs) should be interpreted cautiously as potentially anti-conservative for surveys with design effects (NHIS, MEPS, NHANES).
Show code
# Load combined results from CSVs (avoids memory issues with large RDS files)all_age_effects <-read_csv(file.path(apci_output_dir, "apci_age_effects_all.csv"),show_col_types =FALSE)all_period_effects <-read_csv(file.path(apci_output_dir, "apci_period_effects_all.csv"),show_col_types =FALSE)all_cohort_avgs <-read_csv(file.path(apci_output_dir, "apci_cohort_avgs_all.csv"),show_col_types =FALSE)all_cohort_slopes <-read_csv(file.path(apci_output_dir, "apci_cohort_slopes_all.csv"),show_col_types =FALSE)deviance_summary <-read_csv(file.path(apci_output_dir, "apci_deviance_summary.csv"),show_col_types =FALSE)sv_levels <-toupper(surveys)# Note: Per-survey RDS files are not loaded here to avoid memory issues.# All cross-survey data comes from the combined CSVs above.
2 Summary Table
Show code
deviance_summary |>mutate(`Sig. Cohort Avgs`=paste0(n_sig_avg, "/", n_cohorts),`Sig. Cohort Slopes`=paste0(n_sig_slope, "/", n_cohorts) ) |>select(Survey = survey, `N Cohorts`= n_cohorts,`Sig. Cohort Avgs`, `Sig. Cohort Slopes`) |>kable(caption ="APCI model summary across surveys. Significance at p < 0.05.") |>styled_kable(bootstrap_options =c("striped", "hover", "condensed"),full_width =FALSE)
Table 1: APCI model summary across surveys. Significance at p
Survey
N Cohorts
Sig. Cohort Avgs
Sig. Cohort Slopes
BRFSS
20
11/20
6/20
CPS
20
19/20
15/20
GSS
24
18/24
9/24
MEPS
18
10/18
14/18
NHANES
16
6/16
7/16
NHIS
22
18/22
17/22
3 Cross-Survey Results
3.1 Cohort Deviations (Inter-Cohort)
These plots show the average deviation of each birth cohort from the age + period main effects. Positive values mean the cohort has higher SRH than expected; negative values mean lower SRH than expected.
Figure 1: Inter-cohort deviations from age+period main effects across surveys. Filled points = p < 0.05. Positive = better-than-expected SRH for that cohort.
3.2 Cohort Slopes (Intra-Cohort)
These plots show the intra-cohort life-course slopes. A positive slope means the cohort’s advantage (or disadvantage) accumulates with age; a negative slope means it diminishes; near-zero means it stays constant.
The combined plots for each survey (main effects + cohort averages + cohort slopes) are saved in output/apc/apci/apci_combined_{survey}.png. See those files for detailed per-survey visualizations.
6 Comparison with Other APC Methods
6.1 Median Polish vs APCI
Aspect
Median Polish
APCI
Type
Non-parametric descriptive
Parametric inferential
Identification
Not formally identified
Fully identified (interaction-based)
Hypothesis tests
None (descriptive only)
Global F-test, local z-tests
Cohort concept
Residuals from AP matrix
Age-by-period interaction
Cohort dynamics
Static effect only
Intra-cohort slopes (accumulation/leveling)
Advantages
No distributional assumptions, robust to outliers
Formal inference, richer cohort characterization
Limitations
No CIs or p-values
Assumes GLM; SEs don’t account for survey design
6.2 BHAPC vs APCI
Aspect
BHAPC
APCI
Framework
Bayesian (rstanarm)
Frequentist (glm)
APC effects
Crossed random effects
Fixed main effects + interaction
Identification
Variance decomposition
Interaction = cohort
Cohort concept
Random cohort intercepts
Inter-cohort deviations + intra-cohort slopes
Inference
Posterior distributions
z-tests, F-test
Computation
Slow (MCMC)
Fast (glm)
7 Interpretation for Convergence Paper
The APCI results provide formal statistical evidence for the convergence phenomenon:
Do cohort effects exist? The number of significant cohort deviations across surveys answers this.
Which cohorts are affected? The cohort averages identify which birth cohorts deviate from age+period expectations — recent cohorts (born ~1980+) with negative deviations would support the “declining youth SRH” component of convergence.
Is convergence from both directions? Older cohorts (born ~1930-1950) with positive deviations would confirm the “improving older SRH” component.
Do effects accumulate? Intra-cohort slopes reveal whether cohort advantages/disadvantages grow with age (cumulative advantage theory) or diminish (age-as-leveler).
